Hello François,  these sums are known, a good deal of identities have been discover using Zeta values already.  Also, the S(n+1)-S(n), I don't understand, this is simply the rewritten version of the same formula ?  There is this article of Flajolet and Vardi here : https://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0ahUKEwjB... and this was in 1996, some progress have been made since too. here : https://ac.els-cdn.com/0022247X88900133/1-s2.0-0022247X88900133-main.pdf?_ti... on page 133, you will find your formula. (equivalent). Sorry, but I think this is all known material.  Simon Plouffe Le 2018-03-14 à 14:47, françois mendzina essomba2 a écrit :

sum((-1)^k*(1-1/2^k)*zeta(k), k= 2..n);